The immersion-minimal infinitely edge-connected graph
Paul Knappe, Jan Kurkofka
TL;DR
This paper proves the existence of a unique immersion-minimal infinitely edge-connected graph, the halved Farey graph, and discusses the complexity of minimal lists of such graphs.
Contribution
It establishes the uniqueness of the immersion-minimal infinitely edge-connected graph and explores the uncountability of minimal lists represented as topological minors.
Findings
The halved Farey graph is the unique immersion-minimal infinitely edge-connected graph.
Every infinitely edge-connected graph contains the halved Farey graph as an immersion minor.
Minimal lists of infinitely edge-connected graphs as topological minors are uncountably infinite.
Abstract
We show that there is a unique immersion-minimal infinitely edge-connected graph: every such graph contains the halved Farey graph, which is itself infinitely edge-connected, as an immersion minor. By contrast, any minimal list of infinitely edge-connected graphs represented in all such graphs as topological minors must be uncountable.
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Taxonomy
TopicsAdvanced Graph Theory Research · Computational Geometry and Mesh Generation · Topological and Geometric Data Analysis